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Graphing Linear and Quadratic Functions

It’s that time again folks. Welcome to Beyond’s second Monthly Maths Mastery blog! This month, we’re graphing linear and quadratic functions, while shedding light on some of the key misconceptions. Grab your graph paper, poise those pencils and get set for a masterclass on maths with Beyond.

What is the difference between linear and quadratic functions?

A linear function is one of the form y = mx + c. For each input of x, you get one output for y. The graph of these functions is a single straight line.

A quadratic function is one of the form y = ax2 + bx + c. For each output for y, there can be up to two associated input values of x. The graph of these functions is a parabola – a smooth, approximately u-shaped or n-shaped, curve.

You need to be able to confidently plot the graphs of these functions, and the simplest way to do so is by using a table of values.

How do you draw a graph of a linear function?

Regardless of whether a table is given to you, you should consider using one to ensure you’re correctly graphing linear and quadratic functions.

For example,

Plot the graph of y = 2x – 1 for -3 ≤ x ≤ 3.

This inequality notation means that we should plot the graph for values of x between and including -3 and 3.

A table of values might look as below. Note that if a table of values isn’t given, you can get away with calculating no less than 3 coordinates for a linear function. You might wish to just choose positive values of x for ease!

x

-3

-2

-1

0

1

2

3

y

To find the relevant values for y, substitute the value of x into the equation y = 2x – 1.

When x = -3, y = 2 × (-3) – 1 = -7.

When x = -2, y = 2 × (-2) – 1 = -5.

When x = -1, y = 2 × (-1) – 1 = -3.

The finished table will look like this:

x

-3

-2

-1

0

1

2

3

y

-7

-5

-3

-1

1

3

5

These values are ready to be plotted on a graph. Don’t forget to join them up with a straight line!

Graphing linear functions

At this stage, any points which do not lie on the line have probably been incorrectly calculated. If this is the case, go back and check your working.

Seems okay so far 🤔 suspiciously so…

How do you draw a graph of a quadratic function?

You will be given a table to complete. Once again, to find the values for y, substitute the values of x into the equation given.

For example,

Complete the table of values for the function y = x2 + x – 2.

x

-3

-2

-1

0

1

2

y

4

-2

Remember to be really careful when working with quadratics. We must remember that when we square a negative number, we get a positive result.

When x = -2, y = (-2)2 + (-2) – 2 = 0

When x = -1, y = (-1)2 + (-1) – 2 = -2

When x = 1, y = 12 + 1 – 2 = 0

When x = 2, y = 22 + 2 – 2 = 4

The finished table is:

x

-3

-2

-1

0

1

2

y

4

0

-2

-2

0

4

Quadratic graphs are symmetrical, so the symmetry in our coordinates is a good indication that we’ve calculated these values correctly.

These values are ready to be plotted on a graph. Don’t forget to join them up with a smooth curve!

Graphing quadratic functions

How do you solve equations using a graph?

To solve an equation graphically, draw the graph of each side of the equation as y = f(x) and see where the two graphs intersect…sort of like our tramlines illustration below 😉. The main equation you will need to solve is when it is equal to zero. The line y = 0 is the x-axis, so you need to find the points where your graph crosses the x-axis.

For example,

Use your graph to solve the equation x2 + x – 2 = 0.

We have already drawn the graph for y = x2 + x – 2. The other side of our equation is zero, so we need to think about the line y = 0. This is the x-axis, so look for the points where the graph crosses the x-axis.

The solutions are x = 1 and x = -2.

To solve the equation x2 + x – 2 = 3, we would draw the line y = 3. To solve the equation x2 + x – 2 = x + 1, we would draw the line y = x + 1, and so on!

Does that make sense? It’d be easier if we had a video wouldn’t it? Oh, we do! “Roll VT…”

Graphing linear and quadratic functions? Piece of cake! Same time next month?

In the meantime though, if you’re covering linear and quadratic graphs with your students, check out some of our resources below!